Kronecker. Nuff said. Even the numbers themselves historically started
as positive integers and were subsequently generalized to hell and back.
Here are some other well known concepts that "should" involve $\mathbb{N}$
but were generalized to $\mathbb{Q}$, $\mathbb{R}$ or even $\mathbb{C}$:

Dimension $\rightarrow$ Hausdorff dimension.

Factorial $\rightarrow$ gamma function.

Differentation $\rightarrow$ half-differentation (etc.)

So, can you extend this small to a big list?

(Motivation: Some hypothetic knot polynomial I calculated with
demanded a dimension of its associated group representation
- thus the "rt" tag - of 60/11. That is noooooot boding well
for its existence. :-)

I would rather be interested in the complementary question: what entities, defined on the positive integers, cannot be extended in any sensible way to a larger set?
–
Federico PoloniSep 1 '11 at 13:38

3

Oh, and this should definitely be Community Wiki
–
Federico PoloniSep 1 '11 at 13:39

2

The order of a numerical method in solving $f(x)=0$. At the beginning, it appears to be an integer: $1$ for many methods based upon the Picard contraction principle, $2$ for the Newton--Raphson method. But the order of the secant method is $\phi$, the Gloden ratio!
–
Denis SerreSep 1 '11 at 14:38

7 Answers
7

The writhe is the fundamental differential geometric invariant of a closed space curve. I think it is the most useful topological invariant outside mathematics- biologists use it to study circular DNA molecules, and chemists use it in the study of long polymers. For space curve $C(t)$ it's defined as the double integral
$\frac{1}{4\pi}\int_{C\times C}\frac{C^\prime(s)\times C^\prime(t)\cdot (C(s)-C(t))}{|C(s)-C(t)|^3}ds dt.$

but most people think of it as the number of positive crossings minus the number of negative crossings. This quantity is naturally an integer. The integral formula is based on the Gauss integral for the linking number, but has a complicated history, with a lot of contribution from non-mathematicians.

But, what to do, most real-life long molecules aren't closed space curves. And so biologists, chemists, and physicists, followed by mathematicians, generalized the writhe to open space curves. The idea is that writhe makes sense for a tangle diagram, so they integrated over all projection angles of the open space curve. The result is a definition for the writhe of an open space curve, which is a real number (which can be efficiently estimated). I think it's differential geometry's most useful real numbers for studying open space curves where they occur in biology, chemistry, and physics.

Which is a simultaneous generalization of the integer-valued functions "Euler characteristic" and "count points over finite fields"!
–
JSESep 1 '11 at 16:14

Amen! Also, I guess I bent the rules a little, since measures aren't usually $\mathbf{Z}$-valued, but I think the discrepancy between real numbers and rings of motives is big enough that this still counts. :)
–
David HansenSep 1 '11 at 20:25

Let $f:\mathbb{Z}_p\to\mathbb{Z}_p$ be a "nice" map on the $p$-adic integers (or a map on some more general space with a $p$-adic topology). People who study $p$-adic dynamcis investigate what the iterates of $f$ do to points of the space. So if we fix a point $\alpha\in\mathbb{Z}_p$, we can define an iteration map
$$
I : \mathbb{N} \longrightarrow \mathbb{Z}_p,\qquad
I(n) = f^n(\alpha).
$$
The map $I$ is naturally defined on $\mathbb{N}$, and if $f$ is invertible, then it clearly extends to $\mathbb{Z}$. But for various applications, one would like to evaluate $I(n)$ for $n\in\mathbb{Z}_p$. So the example is

iteration an integral number of times $\to$ iteration a $p$-adic number of times.